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The Operator Product Expansion The Operator Product Expansion Nolan Smyth 6/10/2020 QFT 3 - Final Presentation Outline 1. The Big Picture 2. Renormalization Group Refresher/Setup 3. Introducing the Operator Product Expansion 4. Callen-Symanzik Equation for Wilson Coefficients 5. An example: QCD corrections to weak decays 1 The Big Picture The Big Picture We've spent much of this quarter learning to deal with divergences, typically those of an ultraviolet nature. These divergences led to shifting in mass parameters and coupling constants through the process of renormalization, but were otherwise rather innocuous. Renormalization gave us a great deal of freedom in that we were free to define our theory at any arbitrary momentum scale. But in order to describe the the correct physical theory, the renormalized Green's functions must be identical to the bare Green functions, up to the field strength renormalization. (n) −n=2 (n) GR (µ; x1; :::; xn) = Z Gb (x1; :::; xn) 2 The Big Picture We then asked, how would our Green's functions change if we looked at a different renormalization scale? This led to the Callen-Symanzik (renormalization group) equations. Solving these, we calculated how coupling constants and fields evolve with momentum scale. Until now, we've utilized correlation functions to perform this analysis. But we will see that it is particularly useful to study the renormalization flow at the level of the operators themselves. But to do so, we will need to deal with the non-locality of operator products. This is where we will meet the Operator Product Expansion (OPE). 3 The Big Picture After motivating the OPE, we will see that all of the interesting behavior at high energy (short distances) depends only on Wilson coefficents. These can be calculated once and then used for any process since they do not depend on any external fields that appear in a Green's function. We will then use the OPE to look at QCD renormalization of the weak interactions. Since QCD has asymptotic freedom, we will see that the OPE allows us to determine the renormalization corrections in terms of the anomalous dimensions of the operators. 4 Renormalization Group Refresher/Setup RG The bare and renormalized Green's functions are related by (n) −n=2 (n) GR (µ; x1; :::; xn) = Z Gb (x1; :::; xn) (1) When varying the renormalization scale µ, we must also change Z and the coupling constants (here we'll use λ) in order for these Green's functions to represent the same physical theory. This can be written generally as dG (n) @G (n) @G (n) @λ R = R + R (2) dµ @µ @λ @µ 5 RG By taking the derivative of the left hand side of (1) with respect to µ, and using the fact that the bare Green's function does not depend on the renormalization scale, we find dG (n) n @Z R = − G (n): (3) dµ 2Z @µ R Combining these results, we end up with the renormalization group equation h @ @ i µ + β + nγ G (n) = 0; (4) @µ @λ R @λ µ @Z where β ≡ µ @µ and γ ≡ 2Z @µ control how the coupling constant and field φ vary with µ respectively. 6 RG Correlation functions containing composite operators (operators evaluated at the same spacetime point) must also be renormalized. Ob = ZOOR . Thus, we may write the relationship between the bare and renormalized Green's functions as (n;1) −n=2 −1 (n;1) GR (µ; x1; :::; xn; y) = Z ZO Gb (x1; :::; xn; y); (5) where (n;1) GR (µ; x1; :::; xn; y) = hφ(x1)...φ(xn)O(y)i: (6) 7 RG By the same treatment as before, requiring the bare correlation function to be independent of the renormalization scale µ, we end up with the following equation h @ @ i µ + β + nγ + γ G (n;1) = 0; (7) @µ @λ O R where the anomalous dimension of the composite operator is defined as µ @ZO γO ≡ (8) ZO @µ 8 RG It would be nice to study the renormalization flow at the level of the operators. But the behavior of composite operators is unwieldy (A1(x1)A2(x2):::An(xn) is singular when the arguments coincide (x1 = x2::: = xn)). For example, consider the humble free-field propagator Z d 4p i h0jT φ(x)φ(y)ji = D (x − y) = e−ip·(x−y): (9) F (2π)4 p2 − m2 + i In the limit of x ! y (forming a composite operator), this is divergent! To make our lives easier, we use the OPE to stick all the difficult behavior inside the Wilson coefficients. 9 Introducing the Operator Product Expansion OPE Let's say we're looking at a process that involves two operators, O1 and O2, which act on points separated by a small distance x. Let's also imagine that there are external physical states located much further away φ(yi ). The amplitude for this process can be calculated from the Green's function (where I'm writing all products of operators as time ordered) G12(x; ; y1:::ym) = hO1(x)O2(0)φ(y1)...φ(ym)i: (10) 10 OPE In a correlation function like hO1(x)O2(0)φ(y1)...φ(ym)i, since the local operators φ(yi ) are far away from 0; x, in the limit of x ! 0, the operator product looks like a single local operator. It can, in fact, produce the most general local disturbance. Thus, we can expand it in terms of the basis which contains all possible local operators at 0. 11 OPE The operator product expansion (OPE) states that the product of local operators evaluated at different points, in the limit that those points approach each other, can be written as a sum over composite (local) operators: X lim O1(x)O2(0) = Cn(x)On(0): (11) x!0 n This is useful because it holds at the level of operators; the OPE is independent of external states. In other words, regardless of any fields that might appear in a Green's function, once you have calculated a OPE once, you may then use it to calculate matrix elements for any process. 12 Callen-Symanzik Equation for Wilson Coefficients Wilson Coefficients Using the OPE, our Green's function can be expanded as X n G12(x; y1; :::ym) = C12(x)Gn(y1; :::ym); (12) n where Gn(y1; :::ym) = hOn(0)φ(y1)...φ(ym)i: (13) Notice that, indeed, all of the dependence on x is in the Wilson coefficients (the C(x)'s). But these, in turn, only depend on the operators themselves since X lim O1(x)O2(0) = Cn(x)On(0): (14) x!0 n 13 Wilson Coefficients We expect the Wilson coefficients to depend on the renormalization scale. So to determine this dependence, we write the RG equation for the renormalized Green's function h @ @ i µ + β + nγ + γ + γ G (n) = 0: (15) @µ @λ 1 2 12 The RG equation for the Green functions Gn that show up on the right side of (12) give us h @ @ i µ + β + nγ + γ G = 0; (16) @µ @λ n n where γn is the anomalous dimension of the operator On. 14 Wilson Coefficients In order to satisfy X n G12(x; y1; :::ym) = C12(x)Gn(y1; :::ym); n the Wilson coefficients must obey the equation h @ @ i µ + β + γ + γ − γ C (n) = 0: (17) @µ @λ 1 2 n 12 This shows that our previous assumption of the dependence of the coefficients on µ to be valid. Most importantly, this shows that the Wilson coefficients only depend on the anomalous dimensions of the operators, but not on the specific correlation function! Thus the number of fields φ and their anomalous dimension does not come into play. 15 x-dependence of Wilson Coefficients Let's say the dimensions of the operators O1; O2; and On are d1; d2; and dn respectively. Then we may write the Wilson coefficients as (n) 1 ~n C12 (x; µ) = C12(µjxj); (18) jxjd1+d2−dn ~n where C12 is dimensionless. To find the structure of the coefficient, we introduce the running coupling λ(1=jxj), defined by dλ(p; λ) = β(λ); λ(µ, λ) = λ d ln(p/µ) which lets us write the solution as (n) (n) C12 (λ(1=jxj)) C12 (x; µ) = jxjd1+d2−dn (19) Z µ 0 h dp 0 0 0 i × exp 0 [γn(λ(p )) − γO1 (λ(p )) − γO2 (λ(p ))] 1=jxj p (n) 16 where C12 is a perturbative function of the running coupling. x-dependence of Wilson Coefficients (n) (n) C12 (λ(1=jxj)) C12 (x; µ) = jxjd1+d2−dn Z µ 0 h dp 0 0 0 i × exp 0 [γn(λ(p )) − γO1 (λ(p )) − γO2 (λ(p ))] 1=jxj p We see that the short distance behavior is singular if dn < d1 + d2. On the other hand, if dn > d1 + d2, the contribution goes to zero so we do not need to consider operators of this kind in the OPE. In class, we saw that QCD was asymptotically free. That is, the coupling goes to zero at short distances. Let's examine the Wilson coefficients further in this case. 17 QCD Wilson Coefficients To first order, any anomalous dimension can be written as g 2 γO ≡ −aO (4π)2 for some numerical constant aO. Thus we have α γ − γ − γ = (a + a − a ) s : (20) d 1 2 1 2 n (4π) At one loop, the running coupling αs is simply 4π α (Q2) = (21) s Q2 β0 ln( 2 ) ΛQCD where β0 is the coefficient of the first term in the Taylor expansion of the QCD β function and Q is the momentum scale of the given process.
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